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Bradley Efron

Publications and source records attributed to Bradley Efron.

15 recordsLinked to original sources
arXiv2026

A bootstrap approach to prediction-powered inference

Prediction-powered inference (PPI) refers to a two-level situation where the statistician observes a set of $(x,y)$ pairs and another set of $x$s with the responses $y$ missing. Also available is some independent background data from which a prediction rule $f(x)$ has been produced, perhaps by a machine learning algorithm; $f(x)$ approximates $E\{y\mid x\}$ but there is no guarantee of its accuracy for the situation at hand. Angelopoulos et al. (2023a) developed an algorithm that makes use of all the data, including the unlabeled $x$s, for the estimation of a parameter of interest. A different algorithm is proposed here, using the bootstrap to avoid asymptotics, that is shown to have advantages of efficiency and generality. It is similar in spirit to the original PPI paper by Wang, McCormick and Leek (2020). Prediction-powered inference raises questions about the information available in unlabeled data, with some surprises here, particularly concerning the estimation of the expected value of $y$.

stat.ME↗
arXiv2014

Two Modeling Strategies for Empirical Bayes Estimation

Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_k\sim\mathcal{N}(Θ_k,1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $Θ_k|X_k$. There are two main estimation strategies: modeling on the $θ$ space, called "$g$-modeling" here, and modeling on the $x$ space, called "$f$-modeling." The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.

stat.ME↗
arXiv2014

Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife

We study the variability of predictions made by bagged learners and random forests, and show how to estimate standard errors for these methods. Our work builds on variance estimates for bagging proposed by Efron (1992, 2012) that are based on the jackknife and the infinitesimal jackknife (IJ). In practice, bagged predictors are computed using a finite number B of bootstrap replicates, and working with a large B can be computationally expensive. Direct applications of jackknife and IJ estimators to bagging require B on the order of n^{1.5} bootstrap replicates to converge, where n is the size of the training set. We propose improved versions that only require B on the order of n replicates. Moreover, we show that the IJ estimator requires 1.7 times less bootstrap replicates than the jackknife to achieve a given accuracy. Finally, we study the sampling distributions of the jackknife and IJ variance estimates themselves. We illustrate our findings with multiple experiments and simulation studies.

stat.ML↗
arXiv2013

Bayesian inference and the parametric bootstrap

The parametric bootstrap can be used for the efficient computation of Bayes posterior distributions. Importance sampling formulas take on an easy form relating to the deviance in exponential families and are particularly simple starting from Jeffreys invariant prior. Because of the i.i.d. nature of bootstrap sampling, familiar formulas describe the computational accuracy of the Bayes estimates. Besides computational methods, the theory provides a connection between Bayesian and frequentist analysis. Efficient algorithms for the frequentist accuracy of Bayesian inferences are developed and demonstrated in a model selection example.

stat.AP↗
arXiv2010

The Future of Indirect Evidence

Familiar statistical tests and estimates are obtained by the direct observation of cases of interest: a clinical trial of a new drug, for instance, will compare the drug's effects on a relevant set of patients and controls. Sometimes, though, indirect evidence may be temptingly available, perhaps the results of previous trials on closely related drugs. Very roughly speaking, the difference between direct and indirect statistical evidence marks the boundary between frequentist and Bayesian thinking. Twentieth-century statistical practice focused heavily on direct evidence, on the grounds of superior objectivity. Now, however, new scientific devices such as microarrays routinely produce enormous data sets involving thousands of related situations, where indirect evidence seems too important to ignore. Empirical Bayes methodology offers an attractive direct/indirect compromise. There is already some evidence of a shift toward a less rigid standard of statistical objectivity that allows better use of indirect evidence. This article is basically the text of a recent talk featuring some examples from current practice, with a little bit of futuristic speculation.

stat.ME↗
arXiv2009

Are a set of microarrays independent of each other?

Having observed an $m\times n$ matrix $X$ whose rows are possibly correlated, we wish to test the hypothesis that the columns are independent of each other. Our motivation comes from microarray studies, where the rows of $X$ record expression levels for $m$ different genes, often highly correlated, while the columns represent $n$ individual microarrays, presumably obtained independently. The presumption of independence underlies all the familiar permutation, cross-validation and bootstrap methods for microarray analysis, so it is important to know when independence fails. We develop nonparametric and normal-theory testing methods. The row and column correlations of $X$ interact with each other in a way that complicates test procedures, essentially by reducing the accuracy of the relevant estimators.

stat.AP↗
arXiv2008

Microarrays, Empirical Bayes and the Two-Groups Model

The classic frequentist theory of hypothesis testing developed by Neyman, Pearson and Fisher has a claim to being the twentieth century's most influential piece of applied mathematics. Something new is happening in the twenty-first century: high-throughput devices, such as microarrays, routinely require simultaneous hypothesis tests for thousands of individual cases, not at all what the classical theory had in mind. In these situations empirical Bayes information begins to force itself upon frequentists and Bayesians alike. The two-groups model is a simple Bayesian construction that facilitates empirical Bayes analysis. This article concerns the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focused on Benjamini and Hochberg's False Discovery Rate method. Topics include the choice and meaning of the null hypothesis in large-scale testing situations, power considerations, the limitations of permutation methods, significance testing for groups of cases (such as pathways in microarray studies), correlation effects, multiple confidence intervals and Bayesian competitors to the two-groups model.

stat.ME↗
arXiv2008

Simultaneous inference: When should hypothesis testing problems be combined?

Modern statisticians are often presented with hundreds or thousands of hypothesis testing problems to evaluate at the same time, generated from new scientific technologies such as microarrays, medical and satellite imaging devices, or flow cytometry counters. The relevant statistical literature tends to begin with the tacit assumption that a single combined analysis, for instance, a False Discovery Rate assessment, should be applied to the entire set of problems at hand. This can be a dangerous assumption, as the examples in the paper show, leading to overly conservative or overly liberal conclusions within any particular subclass of the cases. A simple Bayesian theory yields a succinct description of the effects of separation or combination on false discovery rate analyses. The theory allows efficient testing within small subclasses, and has applications to ``enrichment,'' the detection of multi-case effects.

stat.AP↗
arXiv2007

Size, power and false discovery rates

Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr's, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of ``significant'' discoveries.

math.ST↗
arXiv2007

On testing the significance of sets of genes

This paper discusses the problem of identifying differentially expressed groups of genes from a microarray experiment. The groups of genes are externally defined, for example, sets of gene pathways derived from biological databases. Our starting point is the interesting Gene Set Enrichment Analysis (GSEA) procedure of Subramanian et al. [Proc. Natl. Acad. Sci. USA 102 (2005) 15545--15550]. We study the problem in some generality and propose two potential improvements to GSEA: the maxmean statistic for summarizing gene-sets, and restandardization for more accurate inferences. We discuss a variety of examples and extensions, including the use of gene-set scores for class predictions. We also describe a new R language package GSA that implements our ideas.

math.ST↗
arXiv2004

Least Angle Regression

The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method;

math.ST↗
arXiv1998

Nonparametric Methods for Doubly Truncated Data

Truncated data plays an important role in the statistical analysis of astronomical observations as well as in survival analysis. The motivating example for this paper concerns a set of measurements on quasars in which there is double truncation. That is, the quasars are only observed if their luminosity occurs within a certain finite interval, bounded at both ends, with the interval varying for different observations. Nonparametric methods for the testing and estimation of doubly truncated data are developed. These methods extend some known techniques for data that is only truncated on one side, in particular Lynden-Bell's estimator and the truncated version of Kendall's tau statistic. However the kind of hazard function arguments that underlie the one-sided methods fail for two-sided truncation. Bootstrap and Markov Chain Monte Carlo techniques are used here in their place. Finally, we apply these techniques to the quasar data, answering a question about their long-term luminosity evolution.

astro-ph↗